Is there an articulatable difference between the, I don't know, style? or feel? of how each faction plays out? Do the four decks give them different levels of power or thematically different kinds of powers?

There is no difference in the way you play with the 4 factions, the only change are the Event decks. And yes, tematically the decks are different and match every faction. Karkoth (the undead faction) have different cards to raise the dead, Vailin are corsairs and have events to improve the attack from boats, etc.